#356979

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6,11,15,10,14,22,9)(7,16,20,26,19,12,27,18)(8,21,25,23,24,13,29,30)(17,28,34,32,33,31,36,35);(5,7)(6,12,14,16)(8,17)(9,26,15,18)(10,19)(11,20)(13,28,21,31)(22,27)(23,35,30,32)(24,33)(25,34)(29,36);(2,4)(5,8,22,29,10,24,11,25)(6,13,9,23,14,21,15,30)(7,17,27,36,19,33,20,34)(12,28,26,35,16,31,18,32);(5,9,22,14,10,15,11,6)(7,18,27,12,19,26,20,16)(8,23,29,21,24,30,25,13)(17,32,36,28,33,35,34,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fg,g^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,h^d=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1757, #72342938, #72342945, …
maximal normal subgroup of	D₄×#1757, #72342938, #72342945, …
maximal quotient of	#72342997

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264175);